very very near an antenna there is the so called reactive field. The E and H fields are called reactive there because energy is not radiated away.
The fields E and H are 90 degree out of phase (temporally).
The electromagnetic field behaves like a LC tank circuit: energy goes back and forth from the E to the H field and vice versa....

Are the field lines are still attached to the antenna?

Is the reactive field the domain of evanescent waves? Evanescent waves are still waves but propagate at 90 degrees to to the propagation distance and decay exponentially in that same direction....They don't necessarily store energy....Does that mean that they do not correspond to the reactive field?

In the extreme near field, the E- and H- fields are made up of components that die of as 1/r, 1/r^2, and 1/r^3, where r is the distance from the antenna.

The 1/r fields are the radiated fields, they exist even in the near field region.

The 1/r^3 and 1/r^2 fields exist in the near field, and dominate in that region in terms of magnitude. These are 90 degrees out of phase, and do represent the energy stored in the near field.

My understanding of evanescent waves is that they are non-propagting: that is, in stead of varying spatially like exp(i*w*t), they vary as exp(-i*alpha*t). The main example are the fields in a waveguide below the cutoff frequency. Personally, I don't think of the near fields as evanescent, but I suppose that could be argued.

Also, note that in this post, 90 degrees out of phase refers to out of phase time-wise. In the far field, the E- and H- fields are 90 degrees out of phase spatially: that is, if E- is in the +x direction, than H- will be in the y-direction.

By store energy, it's the same as for a capacitor or inductor. When the voltage source is disconnected from a capacitor, the remaining charge on the plates stores the energy, and will bleed off through whatever resistance it sees.

My interpretation is it is the same way for antennas. When the source is disconnected, there is remaining charge on the antenna associated with the reactive fields, which will dissipate in the medium (or flow back down the transmission line if it remains).

The radiated fields carry power away from an antenna and will continue to propagate away from the antenna, even if the source is disconnected.